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NSW Syllabuses

Glossary

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Bernoulli distribution

The Bernoulli distribution is the probability distribution of a random variable which takes the value 1 with ‘success’ probability \(p\), and the value 0 with ‘failure’ probability \(q=1-p\). The Bernoulli distribution is a special case of the binomial distribution, where \(n=1\).

Bernoulli random variable

A Bernoulli random variable has two possible values, namely 0 and 1. The parameter associated with such a random variable is the probability \(p\) of obtaining a 1.

Bernoulli trial

A Bernoulli trial is a chance experiment with only two possible outcomes, typically labelled ‘success’ and ‘failure’.

binomial coefficient

The coefficient of the term \( x^{n-r} y^{r} \) in the expansion of \((x+y)^n\) is called a binomial coefficient. It is written as \(^nC_r\) or \(\binom n r\) where \(r=0,1, \ldots ,n\) and can be found using Pascal’s triangle.

binomial distribution

The binomial distribution with parameters \(n\) and \(p\) is the discrete probability distribution of the number of successes in a sequence of \(n\) independent Bernoulli trials, each of which yields success with probability \(p\).

binomial expansion

A binomial expansion describes the algebraic expansion of powers of a binomial expression, for example, \((x+y)^4\).

binomial random variable

A binomial random variable \(X\) represents the number of successes in \(n\) independent Bernoulli trials. In each Bernoulli trial, the probability of success is \(p\) and the probability of failure is \(q=1-p\).

column vector notation

A vector in two dimensions can be represented in column vector notation. For example, the ordered vector pair \(\left(4,5\right)\) can be represented in column vector notation as \( \begin{bmatrix} 4 \\ 5 \end{bmatrix} \)

combinations

Combinations are selections of \(r\) objects from \(n\) distinct objects, where order is not important. The number of combinations is denoted by \(^n C_r\) or \(\binom nr\), and is given by \(\frac{n!}{r!(n-r)!}\)

component form (of a vector)

The component form of a vector expresses the vector in terms of unit vectors \( \boldsymbol{i} \) (a unit vector in the \(x\)-direction) and \( \boldsymbol{j} \) (a unit vector in the \(y\)-direction). For example, the ordered vector pair \( \left(4, 3\right) \) can be represented as \( 4\boldsymbol{i}  + 3\boldsymbol{j} \)

differential equation

A differential equation is any equation containing the derivative of an unknown function.

direction field

A direction field (or slope field) is a graphical representation of the tangent lines to the solution of a first-order differential equation.

displacement vector

A displacement vector describes the displacement from one point to another. Also known as a relative vector.

double angle formulae

The double angle formulae for trigonometric functions are:

\begin{align*} \sin 2A &= 2\sin A\sin B \\ \cos 2A &= \cos^2A - \sin^2A \\ &=2\cos^2A - 1 \\ &=1-2\sin^2A \\ \tan 2A &= \frac{2\tan A}{1-\tan^2A} \end{align*}

factor theorem

The factor theorem links the factors and zeros of a polynomial.

The factor theorem states that a polynomial \(P(x)\) has a factor \((x-k)\) if and only if \(P(k)=0\), ie \(k\) is a root of the equation \(P(x)=0\).

factorial

The product of the first \(n\) positive integers is called the factorial of \(n\) and is denoted by \(n!\):

\(n! = n(n-1)(n-2)(n-3)\!\!\;\times\cdots\times 3 \times 2 \times 1 \)

For example: \(6! = 6\times5\times4\times3\times2\times1=720\).

fundamental counting principle

The fundamental counting principle states that if one event has \(m\) possible outcomes and a second independent event has \(n\) possible outcomes, then there are \(m\times n\) total possible outcomes for the two combined events.

integrand

An integrand is a function that is to be integrated.

inverse functions

The functions \(f(x)\) and \(g(x)\) are said to be inverse functions if \(f\left(g(x)\right) = x\) and \(g\left(f(x)\right)=x\) for all \(x\) in the specified domain.

The inverse function of \(f(x)\) is notated by the symbol \(f^{-1}(x)\), which is read as ‘\(f\) inverse of \(x\)’.

For example: if \(f(x)=x+1\) and \(g(x)=x-1\), then \(f\left(g(x)\right)=g\left(f(x)\right)=x\) and \(f(x)\) is the inverse function of \(g(x)\) and vice versa.

inverse trigonometric functions

The inverse sine function, \(\boldsymbol y \boldsymbol = \mathbf{arcsin}\, \boldsymbol{x}\)

If the domain for the sine function is restricted to the interval \(\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\), a one-to-one function is formed and so the inverse function exists.

The inverse of this restricted sine function is denoted by \(\arcsin\) or \(\sin^{-1}\) and is defined by:

\(\sin^{-1}:\left[-1,1\right]\rightarrow \mathbb R\). Its range is \( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \).

The inverse cosine function, \(\boldsymbol y \boldsymbol = \mathbf{arccos}\, \boldsymbol{x}\)

If the domain for the cosine function is restricted to the interval \(\left[0,\pi\right]\), a one-to-one function is formed and so the inverse function exists.

The inverse of this restricted cosine function is denoted by \(\arccos\) or \(\cos^{-1}\) and is defined by:

\(\cos^{-1}:\left[-1,1\right]\rightarrow \mathbb R\). Its range is \( \left[0,\pi\right]\).

The inverse tangent function, \(\boldsymbol y \boldsymbol = \mathbf{arctan}\, \boldsymbol{x}\)

If the domain for the tangent function is restricted to the interval \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\), a one-to-one function is formed and so an inverse function exists.

The inverse of this restricted tangent function is denoted by \(\arctan\) or \(\tan^{-1}\) and is defined by:

\(\tan^{-1}:\mathbb R\rightarrow\mathbb R\). Its range is \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\).

logistic equation

The logistic equation is the differential equation \( \frac{dP}{dt} = k P (N-P)\). Thus, if \( P=0\) or \( P = N \), \( \frac{dP}{dt} = 0 \).

magnitude (of a vector)

The magnitude of the vector \( a \boldsymbol i + b \boldsymbol j\)  is the length of the vector and is given by \( \sqrt{a^2+b^2} \).

mathematical induction

Mathematical induction is a method of mathematical proof used to prove propositions (also referred to as statements) involving the natural numbers.

For example: A statement involving the natural number \(n \) is true for every \( n \in \mathbb{N} \) (\( \mathbb{N} \) is the set of all natural numbers) provided that:

  1. The statement is true in the special case \( n = 1 \).
  2. The truth of the statement for \(n = k \), \(k \in \mathbb{N} \) implies the truth of the statement for \(n = k + 1\).

Also known as proof by induction or inductive proof.

The principle of induction is an axiom and so cannot itself be proven.

multiplicity (of a root)

Given a polynomial \(P(x)\), if \(P(x)=(x-a)^k Q(x)\), \( \ Q(a)\ne0\) and \(k>0\), then the root \(x=a\) has multiplicity \(k\).

ordered pairs notation for vectors

A vector in two dimensions can be represented as an ordered pair of numbers, for example \( \left(1,-2\right) \).

parameter

A parameter is a variable in a mathematical expression. It may be thought of either as separate from the dependent and independent variable, for example, \( \theta \) in \( y = x \cos \theta\); or as linking the dependent and independent variable as in \( x = a \cos \theta\), \( y = a \sin \theta\).

Pascal's triangle

Pascal’s triangle is an arrangement of the coefficients in the expansion of \((x+y)^n\) where \(n=1,2,\ldots\). These numbers are presented in a triangle where the \(n\)th row consists of the binomial coefficients \(^nC_r\) or \(\binom nr\), where \(r=0,1,\ldots,n\):

demonstration of pascal's triangle

In Pascal’s triangle any term is the sum of the two terms diagonally ‘above’ it.

For example, \(10=4+6\).

 

permutations

A permutation of \(n\) distinct objects is an arrangement of these \(n\) objects where order is important. The number of permutations of \(n\) objects is \(n!\).

The number of permutations of \(r\) distinct objects chosen from \(n\) distinct objects where order is important, is denoted by \(^nP_r\) and is equal to: \(^nP_r = n(n-1)...(n-r+1) = \frac{n!}{(n-r)!}\)

pigeonhole principle

The pigeonhole principle states that if \(n\) items are put into \(m\) containers, where \(n>m\), then at least one container must contain more than one item.

polar form (of a vector)

The polar form of a vector \(\boldsymbol v\) represents the magnitude \(r\) and direction \(\theta\) (measured from the positive direction of the \(x\)-axis) of the vector as an ordered pair in the form \(\boldsymbol v=\left(r,\theta\right)\).

position vector

The position vector of a point \( P\) in the plane is the vector joining the origin to \( P\).

projectile motion

Projectile motion refers to the motion of an object thrown or projected through the air under gravity. Such motion can be analysed and modelled, with the simplest useful model assuming that:

  • the projectile is a point and has no spin
  • the force due to air resistance is negligible
  • the only force acting on the projectile is the constant force due to gravity, \( g=9.8m/s^2 \), downwards (assuming the projectile is moving close to the Earth’s surface).
quadratic inequality

A quadratic inequality is an inequality involving at least one quadratic expression.

remainder theorem

The remainder theorem states that when dividing a polynomial \(P(x)\) by \((x-k)\), then the remainder is equal to \(P(k)\).

sample proportion

Let there be \(x\) successes out of \(n\) Bernoulli trials. The sample proportion is the fraction of samples which were successes, so \(\hat p= \frac{x}{n}\).

For large \(n\), \(\hat p\) has an approximately normal distribution.

scalar

A scalar is a quantity with magnitude but no direction.

scalar product

The scalar product of vectors \( \boldsymbol a \) and \( \boldsymbol b \) is denoted by \( \boldsymbol a \cdot \boldsymbol b \).

\( \boldsymbol a \cdot \boldsymbol b = x_ 1 x_2 + y_1 y_2\) where \( \boldsymbol a = x_1 \boldsymbol i + y_1 \boldsymbol j\) and \( \boldsymbol b = x_2 \boldsymbol i + y_2 \boldsymbol j \).

\( \boldsymbol a \cdot \boldsymbol b = \lvert \boldsymbol a \rvert \lvert \boldsymbol b \rvert \cos \theta \), where \( \theta \) is the angle between the vectors \( \boldsymbol a \) and \( \boldsymbol b\). 

Also known as the dot product.

solid of revolution

A solid of revolution is formed when a region in the \(xy\) plane is rotated about either the \(x\) or \(y\) axis.

statement

A statement is a sentence that is either true or false. So '3 is an odd integer' is a statement, but '\( \pi \) is a cool number' is not a statement.

sum & difference expansions of trig. functions

The sum and difference expansions for trigonometric functions are:

\(\sin(A\pm B) = \sin A\cos B \pm \sin B\cos A\)

\(\cos(A\pm B) = \cos A\cos B \mp \sin A\sin B\)

\(\tan(A\pm B) = \frac{\tan A\,\pm\,\tan B}{1\,\mp\,\tan A\tan B}\)

 

trigonometric products as sums and differences

Trigonometric products as sums and differences are:

\(\cos A\cos B = \frac{1}{2}\left(\cos(A-B)+\cos(A+B)\right)\)

\(\sin A\sin B = \frac{1}{2}\left(\cos(A-B)-\cos(A+B)\right)\)

\(\sin A\cos B = \frac{1}{2}\left(\sin(A+B)+\sin(A-B)\right)\)

\(\cos A\sin B = \frac{1}{2}\left(\sin(A+B)-\sin(A-B)\right)\)

unit vector

A unit vector is a vector with magnitude 1.

The standard unit vectors are \( \boldsymbol i\) (a unit vector in the \(x\)-direction), and \(\boldsymbol j\) (a unit vector in the \(y\)-direction).

Any non-zero vector \( \underset{^\sim} u \) can be made into a unit vector \( \hat { \underset{^\sim} u} \) by dividing this vector by its length: \( \hat{\underset{^\sim} u} = \frac {\underset{^\sim}u}{\lvert \underset{^\sim}u\rvert}\)

vector

A vector is a quantity that has magnitude and direction.

A vector can be represented using either a bold lower case letter or using a lower case letter with a tilde underneath it. For example, \( \boldsymbol a \) or \( \underset{^\sim}a \).

A vector from point \(A \) to point \(B \) can be represented by \( \overrightarrow{AB} \)